Probability Parameters Identification

Hello all, I recently encountered a seemingly unlikely situation, which compelled me to ponder the odds from a probability perspective. Despite having a math degree, my probability skills are quite rusty and am seeking advice on:
- Identifying needed parameters
- Calculation approach

The situation can be generalized as follows: Let's say you belong to an online community with millions of members worldwide. Within this community, members may choose to stay solo or create (or join) a "Group", which may have up to 50 other members. What are the odds that two (2) members within the same group not only live in the same city but also have a friend in common? While thinking about it, I broke it down as separate probability outcomes:
a. Live in the same town/city
b. Both have a friend in common

I also thought of a 3rd that would be interesting:
c. Know each other in real life

Note: we can assume that which members belong to which group is entirely random. In other words, member interests, geographic region, social status, religion, etc. does not increase the likelihood that they would end up in the same group.

The following are some parameters I was thinking about before throwing in the towel:
1. Total number of community members
2. Total number of community groups
3. Total number of distinct cities lived in by the community members
4. Total population of those cities
5. Population of each city
6. World Population
7. ??

I can't seem to get a mental grasp on how to approach this problem. For example, which parameters would even be needed for calculating the odds of a, b, c? Any tips/guidance is much appreciated! Thank you!

Nate.